On maximal hyperplane sections of the unit ball of $l_p$ for $p>2$

TL;DR

This paper investigates the maximal hyperplane sections of $l_p^n$-balls for $p>2$, extending known results and providing conditions under which these sections resemble those of the cube, with new bounds and counterexamples.

Contribution

It generalizes Ball's cube hyperplane section result to $l_p^n$-balls for a range of $p$, identifying conditions on hyperplane normals and $p$ values for similar maximal sections.

Findings

Maximal hyperplane sections occur for normals with coordinates ≤ 1/√2 under certain $p$ conditions.

Counterexamples exist for large $n$ when normals are aligned with the main diagonal.

Gaussian upper bounds are established for $p$ between 20 and $p_0$.

Abstract

The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p \ge 10^{15}$. By Oleszkiewicz, Ball's result does not transfer to $l_p^n$ for $2 < p < p_0 \simeq 26.265$. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions $n$. We show that the analogue of Ball's result holds in $l_p^n$-balls for all hyperplanes with normal unit vectors $a$, if all coordinates of $a$ have modulus $\le \frac 1 {\sqrt 2}$ and $p$ has distance $\ge 2^{-p}$ to the even integers. Under similar assumptions, we give a Gaussian upper bound for $20 < p < p_0$.